in Replt/ to Professor P. Tardy. 301 



dx 

 Hence the equation (2.) becomes for this instance 



or 



X'(0 + 2r?s(r) = 0. 

 Hence 



In this case the two equations do not determine particular 

 values of a? and y^ the motion not being entirely arbitrary, but 

 such as may arise from the mutual action of the parts of the 

 fluid on each other. While, according to the first equation, 

 the lines of displacement are concentric circles, the second 

 equation shows that the velocity at a given instant varies in- 

 versely as the distance from the centre. 



The " illogical process" by which I have obtained the 

 value of A. in the example in which u = mx and v=- —my, has 

 been employed in several other instances in the same paper, 

 and has always conducted to a correct result. I confess, how- 

 ever, that it contains the same fault as that which Professor 

 Tardy has pointed out in the investigation of equation (1.), 

 viz. that of needlessly introducing the equation (2.). When 

 w, V, w are given explicitly, it is possible to obtain the general 

 equation of the lines of motion, and by consequence the general 

 equation of the surfaces of displacement, by geometrical 

 reasoning. In such instances, therefore, rj/ may be supposed 

 to be given, and A. may be at once inferred from one of the 

 expressions, 



U V w 



w ^' ^' 



dx dy dz 



This, in fact, is what I have virtually done, the equation (2.) 

 not being really made use of. 



Professor Tardy has justly criticized the assumption d&-=. 

 dr=.dr\ This assumption can be made only after proving that 

 the trajectory of the surfaces of displacement is either mo- 

 mentarily or permanently a straight line. On the supposition 

 that udx-\-'vdy-\-'wdz=-(d(^\ it may be shown as follows, by 

 means of the equation (2.), that the orthogonal trajectory is 

 rectilinear. Since udx-\-vdy-\-wdz—{d<^) — 'K{d^)^ it is plain 



